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0024 Weighted DS temporal coherence with noise-free squared coherence weights

Status

Accepted

Date

2026-10-05

Context

The DS temporal coherence ds_temp_coh is |Σₖ exp(jψₖ)| / P over all P image pairs, ψ the residual phase of the phase history. Every pair counts the same, so the incoherent pairs of a point that is coherent only in part of the pairs (short time spans in vegetation, seasons) add random phases and lower it: the point is rejected although its coherent pairs fit. This is the dilution that ISBAS avoids with a hard mask of the pairs (coherence > 0.25).

Under phases that close, ψₖ is estimation noise with variance (1 − |γₖ|²) / (2n|γₖ|²) for n looks (Cramér–Rao bound of the interferometric phase), so the likelihood of independent pairs is Σ κₖ cos ψₖ with κₖ ∝ |γₖ|² / (1 − |γₖ|²); n cancels in the normalization. With the sample coherence the weights are biased: incoherent pairs have E|γ̂|² = 1/n, and at n = 50-121 SHPs thousands of such pairs still carry about half of the weight. Measured with the phases of regularized EMI (decision 0023), 2026-10-04, Xinpu (60 images, 287 000 candidates) and Campi Flegrei (92 images, 400 000):

  • weights |γ̂|²: Xinpu t ≥ 0.8 for 28 % instead of 14 %, but correlation 0.97 with the uniform t, i.e. a shift, not a new ranking;
  • noise-free weights max(0, (|γ̂|² − 1/nₑ) / (1 − 1/nₑ)), nₑ = SHPs / 2: Xinpu 41 %, and a new ranking: among points with uniform t < 0.6, the spatial consistency with reference points 11-20 pixels away (mean cos of the phase difference over the 12-day interferograms) is 0.51 for weighted t ≥ 0.8, 0.28 for 0.6-0.8 and 0.15 below 0.6; selecting the same number of points, the points chosen by the weighted instead of the uniform t have 0.52 instead of 0.35 at the top 40 % (0.71 instead of 0.76 at the top 14 %). Campi Flegrei hardly changes (its coherence is flat in time span).

Decision

  • New ds_temp_coh_weighted(coh, ph, n_looks, image_pairs=None) returns the weighted temporal coherence |Σ wₖ exp(jψₖ)| / Σ wₖ with wₖ = max(0, (|γ̂ₖ|² − 1/n_looks) / (1 − 1/n_looks)) (the squared coherence without the noise bias; exact in mean at γ = 0 and γ = 1; close to the Fisher weight for small coherence and without its divergence at |γ| → 1) and the effective number of image pairs (Σw)² / Σw². NaN and 0 where no pair has a positive weight or n_looks ≤ 1. ds_temp_coh is unchanged. (Changed by decision 0027: ds_temp_coh(..., n_looks=) returns them, ds_temp_coh_weighted is internal.)
  • emperical_co_emi_temp_coh_pc (API weighted, command t_coh_w_dir, eff_n_pairs_dir) computes both with n_looks = number of SHPs / oversampling (pixels per independent look, default 1.0: no assumption, the weaker noise correction). (Changed by decision 0027: always the effective number of looks of the SHP set of decision 0025, no parameter.)
  • The CPU version is numba (one point per iteration); the GPU version is a numba.cuda kernel with one warp per point (coalesced reads of the image pairs, shuffle reduction), as asked by the maintainer: on an A100 100 000 points x 4186 pairs take 5 ms (ds_temp_coh with a thread per point: 22 ms); CPU with 128 threads as fast as ds_temp_coh. numba.cuda kernels take cupy arrays without a copy. The GPU version of ds_temp_coh uses the same kernel structure instead of a cupy kernel with a thread per point (uncoalesced reads): 3.6 ms instead of 22 ms for the same case.

Consequences

  • One parameter, the oversampling of the SLCs; the results depend on it (Xinpu t ≥ 0.8: 34 % for 1, 41 % for 2). It can be measured once per data set (equivalent number of looks of multilooked intensity in homogeneous areas against the number of pixels) instead of guessed; not part of this change.
  • A weighted temporal coherence from few effective pairs is optimistic: select with both outputs.
  • The weighted temporal coherence is not a likelihood test (the pairs of one point are correlated); its expected value for phases that close still depends on the information of the point, so thresholds are empirical.

Do not

  • Do not weight by |γ̂|² without the noise correction: on Xinpu it only shifts the uniform temporal coherence.
  • Do not use the Fisher weights |γ̂|² / (1 − |γ̂|²) without a limit: a few pairs close to 1 dominate.
  • Do not select by the weighted temporal coherence without the effective number of image pairs (Changed by decision 0028: without n_components, not eff_n_pairs.)